The four fraction rules are short: to add or subtract, make the denominators the same and then add or subtract the numerators; to multiply, multiply the tops and multiply the bottoms; to divide, flip the second fraction and multiply. Finish every answer by simplifying it and, if it is top-heavy, turning it into a mixed number.
This cheat sheet gives each rule with worked examples, then covers simplifying, mixed numbers, common mistakes and a 12-question practice set with answers. To check your own working step by step, use the free fraction calculator with steps, and to see why the rules work, line up pieces on the printable fraction strips.
Fraction rules cheat sheet at a glance
| Operation | Fraction rule | Example | Same denominator needed? |
|---|---|---|---|
| Add | Make denominators the same, add numerators, keep the denominator | 1/4 + 2/3 = 3/12 + 8/12 = 11/12 | Yes |
| Subtract | Make denominators the same, subtract numerators, keep the denominator | 5/6 − 1/4 = 10/12 − 3/12 = 7/12 | Yes |
| Multiply | Numerator × numerator, denominator × denominator | 2/3 × 4/5 = 8/15 | No |
| Divide | Keep the first, change ÷ to ×, flip the second | 3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 1 1/2 | No |
How to use this cheat sheet
- Read the sign first. + and − need a common denominator; × and ÷ do not.
- Convert mixed numbers to improper fractions before you start.
- Apply the rule from the table, one step per line.
- Simplify and change top-heavy answers back to mixed numbers.
- Estimate to check. 3/4 ÷ 1/2 should be more than 1, because a half fits into three quarters more than once.
Six words do most of the work. The numerator is the top number and the denominator the bottom. Equivalent fractions are worth the same, such as 1/2 = 2/4. Simplest form means no number except 1 divides both parts. An improper fraction has a top at least as big as its bottom, a mixed number is a whole number plus a fraction, and the reciprocal of 3/4 is 4/3.
Adding fractions with like and unlike denominators
Same denominators
Add the numerators and keep the denominator: 2/7 + 3/7 = 5/7. Two sevenths and three sevenths make five sevenths, just as 2 apples and 3 apples make 5 apples. The denominator names the size of the pieces, so it does not change.
Different denominators
- Find the lowest common multiple (LCM) of the denominators. For 1/4 and 2/3 it is 12.
- Rewrite each fraction over 12: 1/4 = 3/12 and 2/3 = 8/12.
- Add the numerators: 3/12 + 8/12 = 11/12.
- Simplify if you can. 11/12 is already in simplest form.
When the answer is top-heavy, turn it into a mixed number: 5/6 + 3/4 = 10/12 + 9/12 = 19/12 = 1 7/12. Multiplying the two denominators always gives a common denominator, a shortcut shown in the Flinders University rules for fractions, but the LCM keeps the numbers smaller. The LCM calculator finds it for you.
Subtracting fractions and mixed numbers
Subtracting uses the same steps as adding. With equal denominators, subtract the numerators and simplify: 7/8 − 3/8 = 4/8 = 1/2. With different denominators, convert first: 5/6 − 1/4 = 10/12 − 3/12 = 7/12.
For mixed numbers, the safest method is to change both to improper fractions: 3 1/4 − 1 2/3 = 13/4 − 5/3 = 39/12 − 20/12 = 19/12 = 1 7/12. You can also subtract whole numbers and fractions separately, but then you must borrow 1 when the second fraction is bigger, which is where many slips happen.
For written practice, the adding and subtracting fractions worksheets start with like denominators and build up.
Multiplying fractions is the easy one
Multiplying is the easiest of the fraction rules because you need no common denominator: 2/3 × 4/5 = 8/15. If you keep one multiplying fractions sheet on the wall, this is the line to put on it: top times top, bottom times bottom.
Cancel first to keep numbers small
If a numerator and the other fraction’s denominator share a factor, divide both by it before multiplying. In 3/8 × 4/9, 3 and 9 share 3 and 4 and 8 share 4, so the product becomes 1/2 × 1/3 = 1/6. Multiplying first gives 12/72 = 1/6, the same answer with more work.
Whole numbers and mixed numbers
Write a whole number over 1: 4 × 2/3 = 4/1 × 2/3 = 8/3 = 2 2/3. Change mixed numbers to improper fractions before multiplying: 1 1/2 × 2 2/3 = 3/2 × 8/3 = 24/6 = 4. Any multiplying mixed numbers worksheet follows exactly this pattern: convert, multiply, simplify.
Dividing fractions: keep, change, flip
To divide, use keep, change, flip: keep the first fraction, change ÷ to ×, and flip the second fraction to its reciprocal. Then multiply as normal: 5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 = 1 1/4.
Why does flipping work? Dividing asks “how many of these fit into that?” Half a pizza fits into three quarters of a pizza one and a half times, and 3/4 ÷ 1/2 = 3/4 × 2/1 = 3/2 = 1 1/2 agrees. Dividing by a fraction is the same as multiplying by its reciprocal.
- Fraction ÷ whole number: 1/3 ÷ 2 = 1/3 × 1/2 = 1/6
- Whole number ÷ unit fraction: 3 ÷ 1/4 = 3 × 4 = 12, because 12 quarters make 3 wholes.
- Mixed numbers: 2 1/2 ÷ 1 1/4 = 5/2 × 4/5 = 20/10 = 2
A multiplying and dividing fractions worksheet usually mixes both operations, so check the sign before you flip anything. Only the second fraction is flipped, and only when dividing.
Choosing the right operation in word problems
The hardest part of a word problem is choosing the operation. “Altogether” and “more” usually mean adding; “left” and “how much longer” mean subtracting; “of” means multiplying; and “how many pieces fit” or “share equally” mean dividing.
- Of means multiply: a recipe uses 2/3 cup of oats and you make 1 1/2 batches, so 2/3 × 1 1/2 = 2/3 × 3/2 = 1 cup.
- How many fit means divide: a 3/4 m ribbon is cut into 1/8 m pieces, so 3/4 ÷ 1/8 = 3/4 × 8/1 = 6 pieces.
- Left means subtract: a jug holds 1 1/2 litres and 3/4 of a litre is poured out, so 1 1/2 − 3/4 = 6/4 − 3/4 = 3/4 litre is left.
The math word problems worksheets build the same habit: underline the question, pick the fraction rule, then estimate before you calculate.
Simplifying: the last step of every answer
Divide the numerator and denominator by their greatest common factor (GCF). For 18/24 the GCF is 6, so 18/24 = 3/4. If you only spot a smaller factor, divide by it and repeat: 18/24 = 9/12 = 3/4. The simplify fractions calculator shows the GCF, and the equivalent fractions worksheets practise the skill on paper.
Working with mixed numbers
| Task | Method | Example |
|---|---|---|
| Mixed number to improper | Whole × denominator, add numerator, keep denominator | 2 3/5 = 13/5 |
| Improper to mixed number | Divide; the remainder becomes the new numerator | 17/4 = 4 1/4 |
| Add or subtract mixed numbers | Convert to improper fractions, use a common denominator | 1 1/2 + 2 2/3 = 3/2 + 8/3 = 9/6 + 16/6 = 25/6 = 4 1/6 |
| Multiply or divide mixed numbers | Always convert to improper fractions first | 1 1/2 × 2 1/2 = 3/2 × 5/2 = 15/4 = 3 3/4 |
The mixed numbers worksheets practise converting both ways, the step that every rule above relies on.
Common mistakes with fraction rules
| Mistake | Wrong | Right |
|---|---|---|
| Adding the denominators | 1/2 + 1/3 = 2/5 | 1/2 + 1/3 = 5/6 |
| Multiplying whole parts and fractions separately | 1 1/2 × 2 1/2 = 2 1/4 | 1 1/2 × 2 1/2 = 3 3/4 |
| Flipping the first fraction | 2/3 ÷ 4/5 = 3/2 × 4/5 = 6/5 | 2/3 ÷ 4/5 = 2/3 × 5/4 = 5/6 |
| Subtracting the smaller fraction from the bigger one | 3 1/4 − 1 3/4 = 2 2/4 | 3 1/4 − 1 3/4 = 1 1/2 |
| Changing only the denominator | 2/3 = 2/6 | 2/3 = 4/6 |
A quick estimate catches most of these. Half plus a third must be more than a half, so 2/5 cannot be right. Comparing sizes is also good practice in the comparing fractions game.
When children learn each rule, by grade
Children meet these fraction rules over several years. The table follows the Common Core State Standards for Mathematics (US) and the national curriculum in England for key stages 1 and 2.
| Fraction skill | US Common Core | England |
|---|---|---|
| Add and subtract with the same denominator | Grade 4 (4.NF.3), including mixed numbers | Year 3 within one whole; Year 4 beyond |
| Multiply a fraction by a whole number | Grade 4 (4.NF.4) | Year 5, including mixed numbers |
| Add and subtract with different denominators | Grade 5 (5.NF.1) | Year 5 when the denominators are multiples of the same number; Year 6 in general |
| Multiply a fraction by a fraction | Grade 5 (5.NF.4) | Year 6, simple pairs of proper fractions |
| Divide a unit fraction by a whole number, or a whole number by a unit fraction | Grade 5 (5.NF.7) | Year 6: proper fraction ÷ whole number |
| Divide a fraction by a fraction | Grade 6 (6.NS.1) | Not in the primary programme |
The Common Core notes that dividing a fraction by a fraction is not required in grade 5, and England’s Year 6 statement covers only dividing proper fractions by whole numbers.
Practice: 12 questions with answers
Use this as a short multiplying and dividing fractions worksheet with some adding and subtracting mixed in. Give answers in simplest form, using mixed numbers where the answer is bigger than 1.
- 3/10 + 4/10 = ?
- 2/5 + 1/3 = ?
- 7/9 − 1/3 = ?
- 4 1/6 − 2 1/2 = ?
- 5/6 × 3/10 = ?
- 6 × 3/8 = ?
- 2 1/3 × 1 1/2 = ?
- 1 1/4 × 2 2/5 = ?
- 4/5 ÷ 2 = ?
- 3 ÷ 1/4 = ?
- 5/8 ÷ 3/4 = ?
- 3 1/3 ÷ 1 1/9 = ?
Answers: 1) 7/10; 2) 11/15; 3) 4/9; 4) 1 2/3; 5) 1/4; 6) 2 1/4; 7) 3 1/2; 8) 3; 9) 2/5; 10) 12; 11) 5/6; 12) 3.
Questions 5 to 8 make a quick multiplying fractions sheet on their own, and questions 7 and 8 are the kind you meet on any multiplying mixed numbers worksheet. Check any answer in the fraction calculator, which shows each step.
Related guides and printables
- Fractions worksheets from identifying fractions to adding them
- Decimal to fraction calculator
- Multiplication chart for spotting common multiples
- Times tables tricks that speed up cancelling
- Free online math games for kids, including fraction games
Sources
Grade placements: Common Core State Standards for Mathematics, standards 4.NF.3, 4.NF.4, 5.NF.1, 5.NF.4, 5.NF.7 and 6.NS.1; Department for Education, mathematics programmes of study for key stages 1 and 2 (Years 3 to 6, fractions). Common-denominator shortcut: Flinders University Student Learning Centre. This fraction rules cheat sheet was written for PrintSheetLab, and every example on it was checked with Python’s fractions module.
Frequently asked questions
6 quick answers
What are the 4 fraction rules?
Add and subtract with a common denominator, multiply straight across, and divide by flipping the second fraction and multiplying. Then simplify.
Do you need a common denominator to multiply fractions?
No. Multiply the numerators and the denominators: 2/3 × 4/5 = 8/15. Common denominators are only needed to add or subtract.
Why do you flip the second fraction when dividing?
Dividing by a number is the same as multiplying by its reciprocal. For example, 3 ÷ 1/4 = 3 × 4 = 12, because 12 quarters make 3.
How do you add fractions with different denominators?
Rewrite both over the lowest common multiple of the denominators, add the numerators, then simplify: 1/4 + 2/3 = 3/12 + 8/12 = 11/12.
How do the fraction rules work with mixed numbers?
Change mixed numbers to improper fractions, apply the usual rule, then change back: 1 1/2 × 2 2/3 = 3/2 × 8/3 = 4.
What grade learns fraction rules?
In the Common Core, adding like fractions starts in grade 4, unlike denominators and fraction × fraction in grade 5, and fraction ÷ fraction in grade 6.